@dlowan,
See if you can log into the Gravatar account that is linked to your old email.
For that you will probably not need your email to be working. You will probably just need to type in what the email address
was, and whatever the password to the Gravatar account is.
Then save a copy of the graphic so you will always have it if you need it.
If someone else has a copy of your avatar saved, they could either find a way to send you that copy, or they could do a reverse image search to find someplace on the internet that has the image, and give you a link to that place.
If there is still a website somewhere on the internet where you have an account linked to the old address, you could probably get the image off your profile there, and then do a reverse image search to find the full resolution version of the picture.
It might even be stored in all of our browser caches. It's been awhile since I've had any success rummaging through my browser cache, but I could try again later tonight.